On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian manifold and $\mu$ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential $\phi$ there exists a sequence of basic sets $\Omega_n$ such that the topological pressure $P(f|\Omega_n,\phi)$ converges to the free energy $P_{\mu}(\phi) = h(\mu) + \int\phi{d\mu}$. Then we introduce a class of potentials $\phi$ for which there exists sequence of basic sets $\Omega_n$ such that $P(f|\Omega_n,\phi) \to P(\phi)$.